Problem 55 - Thinking Mathematically

Question

Exercises 55-57 present data on a variety of topics. For each data set described in boldface, find the a. mean. b. median. c. mode (or state that there is no mode). d. midrange. $$ \begin{aligned} &\text { 55. Net Worth of the Richest Under } 35\\\ &\begin{array}{|l|c|} \hline \text { Billionaire } & \begin{array}{c} \text { Net Worth } \\ \text { (billions of dollars) } \end{array} \\ \hline \text { John Collison, 26 (Stripe) } & \$ 1.1 \\ \hline \text { Evan Spiegel, 26 (Snap) } & \$ 4.0 \\ \hline \text { Patrick Collison, 28 (Stripe) } & \$ 1.1 \\ \hline \text { Bobby Murphy, 28 (Snap) } & \$ 4.0 \\ \hline \text { Dustin Moskovitz, 32 (Facebook) } & \$ 10.7 \\ \hline \text { Mark Zuckerberg, 32 (Facebook) } & \$ 56.0 \\ \hline \text { Nathan Blecharczyk, 33 (Airbnb) } & \$ 3.8 \\ \hline \text { Ryan Graves, } 33 \text { (Uber) } & \$ 1.6 \\ \hline \text { Kevin Systrom, } 33 \text { (Instagram) } & \$ 1.2 \\ \hline \text { Drew Houston, } 34 \text { (Dropbox) } & \$ 1.0 \\ \hline \text { Liu Ruopeng, } 34 \text { (Technology) } & \$ 1.3 \\ \hline \text { Eduardo Saverin, 34 (Facebook) } & \$ 7.9 \\ \hline \text { Kirill Shamalov, 34 (Petrochemicals) } & \$ 1.3 \\ \hline \text { Evan Sharp, 34 (Pinterest) } & \$ 1.0 \\ \hline \text { Ben Silbermann, 34 (Pinterest) } & \$ 1.6 \\ \hline \end{array} \end{aligned} $$

Step-by-Step Solution

Answer

a. Mean: \$5.75 billion b. Median: \$1.55 billion c. Mode: No mode d. Midrange: \$28.55 billion

Step 1: Calculate the mean

To calculate the mean (average), sum up all the net worth and divide that by the number of billionaires. \[ Mean = \frac{Sum \, of \, Net \, Worths}{Number \, of \, billionaires} \]

Step 2: Calculate the median

To calculate the median, first arrange the data in ascending order. If the number of data points is odd, the median is the middle number. If it's even, the median is the average of the two middle numbers.

Step 3: Find the mode

The mode is the number that appears most frequently. If no number appears more than once, there is no mode.

Step 4: Calculate the midrange

To calculate the midrange, you find the sum of the lowest and highest numbers in the dataset and divide by 2. \[ Midrange = \frac{Lowest \, value + Highest \, value}{2} \]

Key Concepts

Mean

The term 'Mean' is often referred to as the average. It is a way to find the central value of a data set. To calculate it, you sum up all the numbers in your set and then divide by how many numbers there are altogether. For example, if you had a list of numbers, you would add together all the numbers. Then, you would divide that total by the number of items on your list.
This gives you the mean or the average. Mathematically speaking, it’s shown as:

• \( Mean = \frac{Sum \, of \, all \, data \, points}{Total \, number \, of \, data \, points} \)

In the case of the net worths from the exercise, the mean will provide us with the average wealth of the billionaires listed. It serves as a good general measure of the typical net worth in their dataset.

Median

The Median is another type of average, but it’s found differently. The median is the middle value of a dataset arranged in ascending order. It is most useful when you want to identify the "central" value of a set without the influence of outliers or hugely differing values, as can happen with mean.
Here’s how you find it:

• Sort the list of numbers from smallest to largest.
• If there’s an odd number of data points, the median is the number right in the middle.
• If there’s an even number of data points, it’s the average of those middle two numbers.

This makes the median a great choice for understanding what a "typical" data point might be, especially when your data includes some really large (or small) values, like John Collison versus Mark Zuckerberg.

Mode

Mode is the value that appears most frequently in a data set. In some datasets, there may be no mode at all if no number repeats, or there might be multiple modes if several numbers share the highest frequency.
To identify the mode in a dataset:

• List each number along with the count of how often it appears.
• The number with the highest count is your mode.
• If all numbers are unique, you may indicate that the dataset has no mode.

For example, if you are analyzing the net worth, you might find that certain amounts appear more than once, like $1.1 billion and $4.0 billion in our dataset. The one that appears the most is the mode.

Midrange

Midrange is a statistical measure that helps to find the midpoint between the highest and lowest numbers in a dataset. It provides a quick snapshot of the distribution extremes within the data.
Here's how you calculate the midrange:

• Identify the smallest and largest numbers in the data.
• Add them together.
• Divide the sum by two.
• This gives you the midrange.

The formula is:

• \( Midrange = \frac{Lowest \, value + Highest \, value}{2} \)

For our billionaires, comparing the smallest net worth, such as John Collison’s \(1.1 billion, and Mark Zuckerberg’s \)56 billion highlights the spread across the group.